3.1.64 \(\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [64]

Optimal. Leaf size=603 \[ \left (a^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x+\frac {\left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{f}-\frac {d \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{f}+\frac {\left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^3}{3 f}+\frac {\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac {b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f} \]

[Out]

(a^2*(A*c^3-3*A*c*d^2-3*B*c^2*d+B*d^3-C*c^3+3*C*c*d^2)+b^2*(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3-A*(c^3-3*c*d^2))-2
*a*b*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)))*x+(2*a*b*(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3-A*(c^3-3*c*d^2))-a^2*((A
-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2))+b^2*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)))*ln(cos(f*x+e))/f-d*(2*a*b*(c^2*C
+2*B*c*d-C*d^2-A*(c^2-d^2))-a^2*(2*c*(A-C)*d+B*(c^2-d^2))+b^2*(2*c*(A-C)*d+B*(c^2-d^2)))*tan(f*x+e)/f+1/2*(2*a
*b*(A*c-B*d-C*c)+a^2*(B*c+(A-C)*d)-b^2*(B*c+(A-C)*d))*(c+d*tan(f*x+e))^2/f+1/3*(a^2*B-b^2*B+2*a*b*(A-C))*(c+d*
tan(f*x+e))^3/f+1/60*(5*a^2*C*d^2-6*a*b*d*(-5*B*d+C*c)+b^2*(c^2*C-3*B*c*d+15*(A-C)*d^2))*(c+d*tan(f*x+e))^4/d^
3/f-1/15*b*(-3*B*b*d-C*a*d+C*b*c)*tan(f*x+e)*(c+d*tan(f*x+e))^4/d^2/f+1/6*C*(a+b*tan(f*x+e))^2*(c+d*tan(f*x+e)
)^4/d/f

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Rubi [A]
time = 1.08, antiderivative size = 603, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3728, 3718, 3711, 3609, 3606, 3556} \begin {gather*} -\frac {d \tan (e+f x) \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}+\frac {(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{60 d^3 f}+\frac {\log (\cos (e+f x)) \left (-\left (a^2 \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+2 a b \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )+b^2 \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{f}+x \left (a^2 \left (A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right )-2 a b \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )+\frac {\left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^3}{3 f}+\frac {(c+d \tan (e+f x))^2 \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{2 f}-\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]

[Out]

(a^2*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3) + b^2*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3
- A*(c^3 - 3*c*d^2)) - 2*a*b*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*x + ((2*a*b*(c^3*C + 3*B*c^2*d - 3
*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2)) - a^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)) + b^2*((A - C)*d*(3*
c^2 - d^2) + B*(c^3 - 3*c*d^2)))*Log[Cos[e + f*x]])/f - (d*(2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) -
a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Tan[e + f*x])/f + ((2*a*b*(A*c - c*
C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*(c + d*Tan[e + f*x])^2)/(2*f) + ((a^2*B - b^2*B + 2*
a*b*(A - C))*(c + d*Tan[e + f*x])^3)/(3*f) + ((5*a^2*C*d^2 - 6*a*b*d*(c*C - 5*B*d) + b^2*(c^2*C - 3*B*c*d + 15
*(A - C)*d^2))*(c + d*Tan[e + f*x])^4)/(60*d^3*f) - (b*(b*c*C - 3*b*B*d - a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f
*x])^4)/(15*d^2*f) + (C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^4)/(6*d*f)

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3606

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[b*d*(Tan[e + f*x]/f), x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rule 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rule 3718

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])
^(n + 1)/(d*f*(n + 2))), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b
+ a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] &&  !LtQ[n, -1]

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps

\begin {align*} \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}+\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^3 \left (-2 (b c C-3 a A d+2 a C d)+6 (A b+a B-b C) d \tan (e+f x)-2 (b c C-3 b B d-a C d) \tan ^2(e+f x)\right ) \, dx}{6 d}\\ &=-\frac {b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\int (c+d \tan (e+f x))^3 \left (2 \left (6 a b c C d-5 a^2 (3 A-2 C) d^2-b^2 c (c C-3 B d)\right )-30 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)-2 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) \tan ^2(e+f x)\right ) \, dx}{30 d^2}\\ &=\frac {\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac {b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\int (c+d \tan (e+f x))^3 \left (30 \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2-30 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)\right ) \, dx}{30 d^2}\\ &=\frac {\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^3}{3 f}+\frac {\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac {b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\int (c+d \tan (e+f x))^2 \left (-30 d^2 \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )-30 d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)\right ) \, dx}{30 d^2}\\ &=\frac {\left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^3}{3 f}+\frac {\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac {b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\int (c+d \tan (e+f x)) \left (30 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+30 d^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)\right ) \, dx}{30 d^2}\\ &=\left (a^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x-\frac {d \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{f}+\frac {\left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^3}{3 f}+\frac {\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac {b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\left (30 d^3 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+30 c d^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )\right ) \int \tan (e+f x) \, dx}{30 d^2}\\ &=\left (a^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x+\frac {\left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{f}-\frac {d \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{f}+\frac {\left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^3}{3 f}+\frac {\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac {b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.41, size = 419, normalized size = 0.69 \begin {gather*} \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}+\frac {-\frac {2 b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}-\frac {-\frac {\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{2 d f}+\frac {5 \left (3 d \left (2 a b (A c-c C+B d)+a^2 (B c-(A-C) d)-b^2 (B c-(A-C) d)\right ) \left ((i c-d)^3 \log (i-\tan (e+f x))-(i c+d)^3 \log (i+\tan (e+f x))+6 c d^2 \tan (e+f x)+d^3 \tan ^2(e+f x)\right )+\left (a^2 B-b^2 B+2 a b (A-C)\right ) d \left (3 i (c+i d)^4 \log (i-\tan (e+f x))-3 i (c-i d)^4 \log (i+\tan (e+f x))-6 d^2 \left (6 c^2-d^2\right ) \tan (e+f x)-12 c d^3 \tan ^2(e+f x)-2 d^4 \tan ^3(e+f x)\right )\right )}{f}}{5 d}}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]

[Out]

(C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^4)/(6*d*f) + ((-2*b*(b*c*C - 3*b*B*d - a*C*d)*Tan[e + f*x]*(c +
d*Tan[e + f*x])^4)/(5*d*f) - (-1/2*((5*a^2*C*d^2 - 6*a*b*d*(c*C - 5*B*d) + b^2*(c^2*C - 3*B*c*d + 15*(A - C)*
d^2))*(c + d*Tan[e + f*x])^4)/(d*f) + (5*(3*d*(2*a*b*(A*c - c*C + B*d) + a^2*(B*c - (A - C)*d) - b^2*(B*c - (A
- C)*d))*((I*c - d)^3*Log[I - Tan[e + f*x]] - (I*c + d)^3*Log[I + Tan[e + f*x]] + 6*c*d^2*Tan[e + f*x] + d^3*
Tan[e + f*x]^2) + (a^2*B - b^2*B + 2*a*b*(A - C))*d*((3*I)*(c + I*d)^4*Log[I - Tan[e + f*x]] - (3*I)*(c - I*d)
^4*Log[I + Tan[e + f*x]] - 6*d^2*(6*c^2 - d^2)*Tan[e + f*x] - 12*c*d^3*Tan[e + f*x]^2 - 2*d^4*Tan[e + f*x]^3))
)/f)/(5*d))/(6*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1238\) vs. \(2(593)=1186\).
time = 0.36, size = 1239, normalized size = 2.05

method	result	size
norman	\(\left (A \,a^{2} c^{3}-3 A \,a^{2} c \,d^{2}-6 A a b \,c^{2} d +2 A a b \,d^{3}-A \,b^{2} c^{3}+3 A \,b^{2} c \,d^{2}-3 B \,a^{2} c^{2} d +B \,a^{2} d^{3}-2 B a b \,c^{3}+6 B a b c \,d^{2}+3 B \,b^{2} c^{2} d -B \,b^{2} d^{3}-C \,a^{2} c^{3}+3 C \,a^{2} c \,d^{2}+6 C a b \,c^{2} d -2 C a b \,d^{3}+C \,b^{2} c^{3}-3 C \,b^{2} c \,d^{2}\right ) x +\frac {\left (3 A \,a^{2} c \,d^{2}+6 A a b \,c^{2} d -2 A a b \,d^{3}+A \,b^{2} c^{3}-3 A \,b^{2} c \,d^{2}+3 B \,a^{2} c^{2} d -B \,a^{2} d^{3}+2 B a b \,c^{3}-6 B a b c \,d^{2}-3 B \,b^{2} c^{2} d +B \,b^{2} d^{3}+C \,a^{2} c^{3}-3 C \,a^{2} c \,d^{2}-6 C a b \,c^{2} d +2 C a b \,d^{3}-C \,b^{2} c^{3}+3 C \,b^{2} c \,d^{2}\right ) \tan \left (f x +e \right )}{f}+\frac {\left (2 A a b \,d^{3}+3 A \,b^{2} c \,d^{2}+B \,a^{2} d^{3}+6 B a b c \,d^{2}+3 B \,b^{2} c^{2} d -B \,b^{2} d^{3}+3 C \,a^{2} c \,d^{2}+6 C a b \,c^{2} d -2 C a b \,d^{3}+C \,b^{2} c^{3}-3 C \,b^{2} c \,d^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {\left (A \,a^{2} d^{3}+6 A a b c \,d^{2}+3 A \,b^{2} c^{2} d -A \,b^{2} d^{3}+3 B \,a^{2} c \,d^{2}+6 B a b \,c^{2} d -2 B a b \,d^{3}+B \,b^{2} c^{3}-3 B \,b^{2} c \,d^{2}+3 C \,a^{2} c^{2} d -C \,a^{2} d^{3}+2 C a b \,c^{3}-6 C a b c \,d^{2}-3 C \,b^{2} c^{2} d +C \,b^{2} d^{3}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {d \left (A \,b^{2} d^{2}+2 B a b \,d^{2}+3 B \,b^{2} c d +a^{2} C \,d^{2}+6 C a b c d +3 C \,b^{2} c^{2}-C \,b^{2} d^{2}\right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}+\frac {C \,b^{2} d^{3} \left (\tan ^{6}\left (f x +e \right )\right )}{6 f}+\frac {b \,d^{2} \left (B b d +2 a C d +3 C b c \right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5 f}+\frac {\left (3 A \,a^{2} c^{2} d -A \,a^{2} d^{3}+2 A a b \,c^{3}-6 A a b c \,d^{2}-3 A \,b^{2} c^{2} d +A \,b^{2} d^{3}+B \,a^{2} c^{3}-3 B \,a^{2} c \,d^{2}-6 B a b \,c^{2} d +2 B a b \,d^{3}-B \,b^{2} c^{3}+3 B \,b^{2} c \,d^{2}-3 C \,a^{2} c^{2} d +C \,a^{2} d^{3}-2 C a b \,c^{3}+6 C a b c \,d^{2}+3 C \,b^{2} c^{2} d -C \,b^{2} d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}\)	\(896\)
derivativedivides	\(\text {Expression too large to display}\)	\(1239\)
default	\(\text {Expression too large to display}\)	\(1239\)
risch	\(\text {Expression too large to display}\)	\(4231\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x,method=_RETURNVERBOSE)

[Out]

1/f*(3*A*a*b*c*d^2*tan(f*x+e)^2+3*B*a*b*c^2*d*tan(f*x+e)^2+A*b^2*c*d^2*tan(f*x+e)^3+B*b^2*c^2*d*tan(f*x+e)^3+C
*a^2*c*d^2*tan(f*x+e)^3+C*a*b*c^3*tan(f*x+e)^2+3/2*B*a^2*c*d^2*tan(f*x+e)^2-B*a*b*d^3*tan(f*x+e)^2-3/2*B*b^2*c
*d^2*tan(f*x+e)^2+3/2*C*a^2*c^2*d*tan(f*x+e)^2-3/2*C*b^2*c^2*d*tan(f*x+e)^2+3/5*C*b^2*c*d^2*tan(f*x+e)^5+1/2*B
*a*b*d^3*tan(f*x+e)^4+3/4*B*b^2*c*d^2*tan(f*x+e)^4+3/4*C*b^2*c^2*d*tan(f*x+e)^4+2/3*A*a*b*d^3*tan(f*x+e)^3-2/3
*C*a*b*d^3*tan(f*x+e)^3-C*b^2*c*d^2*tan(f*x+e)^3+3/2*A*b^2*c^2*d*tan(f*x+e)^2+3*B*a^2*c^2*d*tan(f*x+e)+2*B*a*b
*c^3*tan(f*x+e)-3*B*b^2*c^2*d*tan(f*x+e)-3*C*a^2*c*d^2*tan(f*x+e)+2*C*a*b*d^3*tan(f*x+e)+3*C*b^2*c*d^2*tan(f*x
+e)+3*A*a^2*c*d^2*tan(f*x+e)-2*A*a*b*d^3*tan(f*x+e)-3*A*b^2*c*d^2*tan(f*x+e)+2/5*C*a*b*d^3*tan(f*x+e)^5+1/2*(3
*A*a^2*c^2*d-A*a^2*d^3+2*A*a*b*c^3-6*A*a*b*c*d^2-3*A*b^2*c^2*d+A*b^2*d^3+B*a^2*c^3-3*B*a^2*c*d^2-6*B*a*b*c^2*d
+2*B*a*b*d^3-B*b^2*c^3+3*B*b^2*c*d^2-3*C*a^2*c^2*d+C*a^2*d^3-2*C*a*b*c^3+6*C*a*b*c*d^2+3*C*b^2*c^2*d-C*b^2*d^3
)*ln(1+tan(f*x+e)^2)+(A*a^2*c^3-3*A*a^2*c*d^2-6*A*a*b*c^2*d+2*A*a*b*d^3-A*b^2*c^3+3*A*b^2*c*d^2-3*B*a^2*c^2*d+
B*a^2*d^3-2*B*a*b*c^3+6*B*a*b*c*d^2+3*B*b^2*c^2*d-B*b^2*d^3-C*a^2*c^3+3*C*a^2*c*d^2+6*C*a*b*c^2*d-2*C*a*b*d^3+
C*b^2*c^3-3*C*b^2*c*d^2)*arctan(tan(f*x+e))+1/2*C*b^2*d^3*tan(f*x+e)^2+1/2*A*a^2*d^3*tan(f*x+e)^2-1/2*A*b^2*d^
3*tan(f*x+e)^2+1/2*B*b^2*c^3*tan(f*x+e)^2-B*a^2*d^3*tan(f*x+e)+B*b^2*d^3*tan(f*x+e)+C*a^2*c^3*tan(f*x+e)-C*b^2
*c^3*tan(f*x+e)+A*b^2*c^3*tan(f*x+e)+1/5*B*b^2*d^3*tan(f*x+e)^5+1/6*C*b^2*d^3*tan(f*x+e)^6+1/4*A*b^2*d^3*tan(f
*x+e)^4+1/4*C*a^2*d^3*tan(f*x+e)^4-1/4*C*b^2*d^3*tan(f*x+e)^4+1/3*B*a^2*d^3*tan(f*x+e)^3-1/3*B*b^2*d^3*tan(f*x
+e)^3+1/3*C*b^2*c^3*tan(f*x+e)^3-1/2*C*a^2*d^3*tan(f*x+e)^2-3*C*a*b*c*d^2*tan(f*x+e)^2+3/2*C*a*b*c*d^2*tan(f*x
+e)^4+2*B*a*b*c*d^2*tan(f*x+e)^3+2*C*a*b*c^2*d*tan(f*x+e)^3+6*A*a*b*c^2*d*tan(f*x+e)-6*B*a*b*c*d^2*tan(f*x+e)-
6*C*a*b*c^2*d*tan(f*x+e))

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Maxima [A]
time = 0.52, size = 688, normalized size = 1.14 \begin {gather*} \frac {10 \, C b^{2} d^{3} \tan \left (f x + e\right )^{6} + 12 \, {\left (3 \, C b^{2} c d^{2} + {\left (2 \, C a b + B b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{5} + 15 \, {\left (3 \, C b^{2} c^{2} d + 3 \, {\left (2 \, C a b + B b^{2}\right )} c d^{2} + {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left (C b^{2} c^{3} + 3 \, {\left (2 \, C a b + B b^{2}\right )} c^{2} d + 3 \, {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c d^{2} + {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{3} + 30 \, {\left ({\left (2 \, C a b + B b^{2}\right )} c^{3} + 3 \, {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c^{2} d + 3 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d^{2} + {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} + 60 \, {\left ({\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c^{3} - 3 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c d^{2} + {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{3}\right )} {\left (f x + e\right )} + 30 \, {\left ({\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{3} + 3 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c^{2} d - 3 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d^{2} - {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 60 \, {\left ({\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c^{3} + 3 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{2} d + 3 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c d^{2} - {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )}{60 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

1/60*(10*C*b^2*d^3*tan(f*x + e)^6 + 12*(3*C*b^2*c*d^2 + (2*C*a*b + B*b^2)*d^3)*tan(f*x + e)^5 + 15*(3*C*b^2*c^
2*d + 3*(2*C*a*b + B*b^2)*c*d^2 + (C*a^2 + 2*B*a*b + (A - C)*b^2)*d^3)*tan(f*x + e)^4 + 20*(C*b^2*c^3 + 3*(2*C
*a*b + B*b^2)*c^2*d + 3*(C*a^2 + 2*B*a*b + (A - C)*b^2)*c*d^2 + (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*tan(f*x +
e)^3 + 30*((2*C*a*b + B*b^2)*c^3 + 3*(C*a^2 + 2*B*a*b + (A - C)*b^2)*c^2*d + 3*(B*a^2 + 2*(A - C)*a*b - B*b^2
)*c*d^2 + ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*d^3)*tan(f*x + e)^2 + 60*(((A - C)*a^2 - 2*B*a*b - (A - C)*b^2
)*c^3 - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c^2*d - 3*((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 + (B*a^2 + 2*(
A - C)*a*b - B*b^2)*d^3)*(f*x + e) + 30*((B*a^2 + 2*(A - C)*a*b - B*b^2)*c^3 + 3*((A - C)*a^2 - 2*B*a*b - (A -
C)*b^2)*c^2*d - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c*d^2 - ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*d^3)*log(tan(
f*x + e)^2 + 1) + 60*((C*a^2 + 2*B*a*b + (A - C)*b^2)*c^3 + 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c^2*d + 3*((A -
C)*a^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 - (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*tan(f*x + e))/f

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Fricas [A]
time = 2.45, size = 686, normalized size = 1.14 \begin {gather*} \frac {10 \, C b^{2} d^{3} \tan \left (f x + e\right )^{6} + 12 \, {\left (3 \, C b^{2} c d^{2} + {\left (2 \, C a b + B b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{5} + 15 \, {\left (3 \, C b^{2} c^{2} d + 3 \, {\left (2 \, C a b + B b^{2}\right )} c d^{2} + {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left (C b^{2} c^{3} + 3 \, {\left (2 \, C a b + B b^{2}\right )} c^{2} d + 3 \, {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c d^{2} + {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{3} + 60 \, {\left ({\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c^{3} - 3 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c d^{2} + {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{3}\right )} f x + 30 \, {\left ({\left (2 \, C a b + B b^{2}\right )} c^{3} + 3 \, {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c^{2} d + 3 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d^{2} + {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} - 30 \, {\left ({\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{3} + 3 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c^{2} d - 3 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c d^{2} - {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d^{3}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 60 \, {\left ({\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c^{3} + 3 \, {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{2} d + 3 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c d^{2} - {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )}{60 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

1/60*(10*C*b^2*d^3*tan(f*x + e)^6 + 12*(3*C*b^2*c*d^2 + (2*C*a*b + B*b^2)*d^3)*tan(f*x + e)^5 + 15*(3*C*b^2*c^
2*d + 3*(2*C*a*b + B*b^2)*c*d^2 + (C*a^2 + 2*B*a*b + (A - C)*b^2)*d^3)*tan(f*x + e)^4 + 20*(C*b^2*c^3 + 3*(2*C
*a*b + B*b^2)*c^2*d + 3*(C*a^2 + 2*B*a*b + (A - C)*b^2)*c*d^2 + (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*tan(f*x +
e)^3 + 60*(((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c^3 - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c^2*d - 3*((A - C)*a
^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 + (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*f*x + 30*((2*C*a*b + B*b^2)*c^3 + 3*(
C*a^2 + 2*B*a*b + (A - C)*b^2)*c^2*d + 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c*d^2 + ((A - C)*a^2 - 2*B*a*b - (A -
C)*b^2)*d^3)*tan(f*x + e)^2 - 30*((B*a^2 + 2*(A - C)*a*b - B*b^2)*c^3 + 3*((A - C)*a^2 - 2*B*a*b - (A - C)*b^
2)*c^2*d - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c*d^2 - ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*d^3)*log(1/(tan(f*x
+ e)^2 + 1)) + 60*((C*a^2 + 2*B*a*b + (A - C)*b^2)*c^3 + 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c^2*d + 3*((A - C)
*a^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 - (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*tan(f*x + e))/f

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1819 vs. \(2 (547) = 1094\).
time = 0.49, size = 1819, normalized size = 3.02 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))**2*(c+d*tan(f*x+e))**3*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

Piecewise((A*a**2*c**3*x + 3*A*a**2*c**2*d*log(tan(e + f*x)**2 + 1)/(2*f) - 3*A*a**2*c*d**2*x + 3*A*a**2*c*d**
2*tan(e + f*x)/f - A*a**2*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + A*a**2*d**3*tan(e + f*x)**2/(2*f) + A*a*b*c**3
*log(tan(e + f*x)**2 + 1)/f - 6*A*a*b*c**2*d*x + 6*A*a*b*c**2*d*tan(e + f*x)/f - 3*A*a*b*c*d**2*log(tan(e + f*
x)**2 + 1)/f + 3*A*a*b*c*d**2*tan(e + f*x)**2/f + 2*A*a*b*d**3*x + 2*A*a*b*d**3*tan(e + f*x)**3/(3*f) - 2*A*a*
b*d**3*tan(e + f*x)/f - A*b**2*c**3*x + A*b**2*c**3*tan(e + f*x)/f - 3*A*b**2*c**2*d*log(tan(e + f*x)**2 + 1)/
(2*f) + 3*A*b**2*c**2*d*tan(e + f*x)**2/(2*f) + 3*A*b**2*c*d**2*x + A*b**2*c*d**2*tan(e + f*x)**3/f - 3*A*b**2
*c*d**2*tan(e + f*x)/f + A*b**2*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + A*b**2*d**3*tan(e + f*x)**4/(4*f) - A*b*
*2*d**3*tan(e + f*x)**2/(2*f) + B*a**2*c**3*log(tan(e + f*x)**2 + 1)/(2*f) - 3*B*a**2*c**2*d*x + 3*B*a**2*c**2
*d*tan(e + f*x)/f - 3*B*a**2*c*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*a**2*c*d**2*tan(e + f*x)**2/(2*f) + B
*a**2*d**3*x + B*a**2*d**3*tan(e + f*x)**3/(3*f) - B*a**2*d**3*tan(e + f*x)/f - 2*B*a*b*c**3*x + 2*B*a*b*c**3*
tan(e + f*x)/f - 3*B*a*b*c**2*d*log(tan(e + f*x)**2 + 1)/f + 3*B*a*b*c**2*d*tan(e + f*x)**2/f + 6*B*a*b*c*d**2
*x + 2*B*a*b*c*d**2*tan(e + f*x)**3/f - 6*B*a*b*c*d**2*tan(e + f*x)/f + B*a*b*d**3*log(tan(e + f*x)**2 + 1)/f
+ B*a*b*d**3*tan(e + f*x)**4/(2*f) - B*a*b*d**3*tan(e + f*x)**2/f - B*b**2*c**3*log(tan(e + f*x)**2 + 1)/(2*f)
+ B*b**2*c**3*tan(e + f*x)**2/(2*f) + 3*B*b**2*c**2*d*x + B*b**2*c**2*d*tan(e + f*x)**3/f - 3*B*b**2*c**2*d*t
an(e + f*x)/f + 3*B*b**2*c*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*b**2*c*d**2*tan(e + f*x)**4/(4*f) - 3*B*b
**2*c*d**2*tan(e + f*x)**2/(2*f) - B*b**2*d**3*x + B*b**2*d**3*tan(e + f*x)**5/(5*f) - B*b**2*d**3*tan(e + f*x
)**3/(3*f) + B*b**2*d**3*tan(e + f*x)/f - C*a**2*c**3*x + C*a**2*c**3*tan(e + f*x)/f - 3*C*a**2*c**2*d*log(tan
(e + f*x)**2 + 1)/(2*f) + 3*C*a**2*c**2*d*tan(e + f*x)**2/(2*f) + 3*C*a**2*c*d**2*x + C*a**2*c*d**2*tan(e + f*
x)**3/f - 3*C*a**2*c*d**2*tan(e + f*x)/f + C*a**2*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + C*a**2*d**3*tan(e + f*
x)**4/(4*f) - C*a**2*d**3*tan(e + f*x)**2/(2*f) - C*a*b*c**3*log(tan(e + f*x)**2 + 1)/f + C*a*b*c**3*tan(e + f
*x)**2/f + 6*C*a*b*c**2*d*x + 2*C*a*b*c**2*d*tan(e + f*x)**3/f - 6*C*a*b*c**2*d*tan(e + f*x)/f + 3*C*a*b*c*d**
2*log(tan(e + f*x)**2 + 1)/f + 3*C*a*b*c*d**2*tan(e + f*x)**4/(2*f) - 3*C*a*b*c*d**2*tan(e + f*x)**2/f - 2*C*a
*b*d**3*x + 2*C*a*b*d**3*tan(e + f*x)**5/(5*f) - 2*C*a*b*d**3*tan(e + f*x)**3/(3*f) + 2*C*a*b*d**3*tan(e + f*x
)/f + C*b**2*c**3*x + C*b**2*c**3*tan(e + f*x)**3/(3*f) - C*b**2*c**3*tan(e + f*x)/f + 3*C*b**2*c**2*d*log(tan
(e + f*x)**2 + 1)/(2*f) + 3*C*b**2*c**2*d*tan(e + f*x)**4/(4*f) - 3*C*b**2*c**2*d*tan(e + f*x)**2/(2*f) - 3*C*
b**2*c*d**2*x + 3*C*b**2*c*d**2*tan(e + f*x)**5/(5*f) - C*b**2*c*d**2*tan(e + f*x)**3/f + 3*C*b**2*c*d**2*tan(
e + f*x)/f - C*b**2*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + C*b**2*d**3*tan(e + f*x)**6/(6*f) - C*b**2*d**3*tan(
e + f*x)**4/(4*f) + C*b**2*d**3*tan(e + f*x)**2/(2*f), Ne(f, 0)), (x*(a + b*tan(e))**2*(c + d*tan(e))**3*(A +
B*tan(e) + C*tan(e)**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 24014 vs. \(2 (602) = 1204\).
time = 20.71, size = 24014, normalized size = 39.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="giac")

[Out]

1/60*(60*A*a^2*c^3*f*x*tan(f*x)^6*tan(e)^6 - 60*C*a^2*c^3*f*x*tan(f*x)^6*tan(e)^6 - 120*B*a*b*c^3*f*x*tan(f*x)
^6*tan(e)^6 - 60*A*b^2*c^3*f*x*tan(f*x)^6*tan(e)^6 + 60*C*b^2*c^3*f*x*tan(f*x)^6*tan(e)^6 - 180*B*a^2*c^2*d*f*
x*tan(f*x)^6*tan(e)^6 - 360*A*a*b*c^2*d*f*x*tan(f*x)^6*tan(e)^6 + 360*C*a*b*c^2*d*f*x*tan(f*x)^6*tan(e)^6 + 18
0*B*b^2*c^2*d*f*x*tan(f*x)^6*tan(e)^6 - 180*A*a^2*c*d^2*f*x*tan(f*x)^6*tan(e)^6 + 180*C*a^2*c*d^2*f*x*tan(f*x)
^6*tan(e)^6 + 360*B*a*b*c*d^2*f*x*tan(f*x)^6*tan(e)^6 + 180*A*b^2*c*d^2*f*x*tan(f*x)^6*tan(e)^6 - 180*C*b^2*c*
d^2*f*x*tan(f*x)^6*tan(e)^6 + 60*B*a^2*d^3*f*x*tan(f*x)^6*tan(e)^6 + 120*A*a*b*d^3*f*x*tan(f*x)^6*tan(e)^6 - 1
20*C*a*b*d^3*f*x*tan(f*x)^6*tan(e)^6 - 60*B*b^2*d^3*f*x*tan(f*x)^6*tan(e)^6 - 30*B*a^2*c^3*log(4*(tan(f*x)^4*t
an(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(
f*x)^6*tan(e)^6 - 60*A*a*b*c^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*
x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 + 60*C*a*b*c^3*log(4*(tan(f*x)^4*tan(e)^2 -
2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan
(e)^6 + 30*B*b^2*c^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*t
an(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 - 90*A*a^2*c^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*
x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 +
90*C*a^2*c^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x
)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 + 180*B*a*b*c^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*
tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 + 90*A*
b^2*c^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan
(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 - 90*C*b^2*c^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e)
+ tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 + 90*B*a^2*c*
d^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) +
1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 + 180*A*a*b*c*d^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + ta
n(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 - 180*C*a*b*c*d^2*
log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(
tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 - 90*B*b^2*c*d^2*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x
)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 + 30*A*a^2*d^3*log(4*(t
an(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2
+ 1))*tan(f*x)^6*tan(e)^6 - 30*C*a^2*d^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)
^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 - 60*B*a*b*d^3*log(4*(tan(f*x)^4*
tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan
(f*x)^6*tan(e)^6 - 30*A*b^2*d^3*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f
*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*tan(e)^6 + 30*C*b^2*d^3*log(4*(tan(f*x)^4*tan(e)^2 -
2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^6*ta
n(e)^6 - 360*A*a^2*c^3*f*x*tan(f*x)^5*tan(e)^5 + 360*C*a^2*c^3*f*x*tan(f*x)^5*tan(e)^5 + 720*B*a*b*c^3*f*x*tan
(f*x)^5*tan(e)^5 + 360*A*b^2*c^3*f*x*tan(f*x)^5*tan(e)^5 - 360*C*b^2*c^3*f*x*tan(f*x)^5*tan(e)^5 + 1080*B*a^2*
c^2*d*f*x*tan(f*x)^5*tan(e)^5 + 2160*A*a*b*c^2*d*f*x*tan(f*x)^5*tan(e)^5 - 2160*C*a*b*c^2*d*f*x*tan(f*x)^5*tan
(e)^5 - 1080*B*b^2*c^2*d*f*x*tan(f*x)^5*tan(e)^5 + 1080*A*a^2*c*d^2*f*x*tan(f*x)^5*tan(e)^5 - 1080*C*a^2*c*d^2
*f*x*tan(f*x)^5*tan(e)^5 - 2160*B*a*b*c*d^2*f*x*tan(f*x)^5*tan(e)^5 - 1080*A*b^2*c*d^2*f*x*tan(f*x)^5*tan(e)^5
+ 1080*C*b^2*c*d^2*f*x*tan(f*x)^5*tan(e)^5 - 360*B*a^2*d^3*f*x*tan(f*x)^5*tan(e)^5 - 720*A*a*b*d^3*f*x*tan(f*
x)^5*tan(e)^5 + 720*C*a*b*d^3*f*x*tan(f*x)^5*tan(e)^5 + 360*B*b^2*d^3*f*x*tan(f*x)^5*tan(e)^5 + 60*C*a*b*c^3*t
an(f*x)^6*tan(e)^6 + 30*B*b^2*c^3*tan(f*x)^6*tan(e)^6 + 90*C*a^2*c^2*d*tan(f*x)^6*tan(e)^6 + 180*B*a*b*c^2*d*t
an(f*x)^6*tan(e)^6 + 90*A*b^2*c^2*d*tan(f*x)^6*tan(e)^6 - 135*C*b^2*c^2*d*tan(f*x)^6*tan(e)^6 + 90*B*a^2*c*d^2
*tan(f*x)^6*tan(e)^6 + 180*A*a*b*c*d^2*tan(f*x)^6*tan(e)^6 - 270*C*a*b*c*d^2*tan(f*x)^6*tan(e)^6 - 135*B*b^2*c
*d^2*tan(f*x)^6*tan(e)^6 + 30*A*a^2*d^3*tan(f*x)^6*tan(e)^6 - 45*C*a^2*d^3*tan(f*x)^6*tan(e)^6 - 90*B*a*b*d^3*
tan(f*x)^6*tan(e)^6 - 45*A*b^2*d^3*tan(f*x)^6*tan(e)^6 + 55*C*b^2*d^3*tan(f*x)^6*tan(e)^6 + 180*B*a^2*c^3*log(
4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + ...

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Mupad [B]
time = 9.31, size = 891, normalized size = 1.48 \begin {gather*} x\,\left (A\,a^2\,c^3-A\,b^2\,c^3+B\,a^2\,d^3-C\,a^2\,c^3-B\,b^2\,d^3+C\,b^2\,c^3+2\,A\,a\,b\,d^3-2\,B\,a\,b\,c^3-2\,C\,a\,b\,d^3-3\,A\,a^2\,c\,d^2+3\,A\,b^2\,c\,d^2-3\,B\,a^2\,c^2\,d+3\,B\,b^2\,c^2\,d+3\,C\,a^2\,c\,d^2-3\,C\,b^2\,c\,d^2-6\,A\,a\,b\,c^2\,d+6\,B\,a\,b\,c\,d^2+6\,C\,a\,b\,c^2\,d\right )-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,a^2\,d^3-A\,b^2\,c^3-b\,d^2\,\left (B\,b\,d+2\,C\,a\,d+3\,C\,b\,c\right )-C\,a^2\,c^3+C\,b^2\,c^3+2\,A\,a\,b\,d^3-2\,B\,a\,b\,c^3-3\,A\,a^2\,c\,d^2+3\,A\,b^2\,c\,d^2-3\,B\,a^2\,c^2\,d+3\,B\,b^2\,c^2\,d+3\,C\,a^2\,c\,d^2-6\,A\,a\,b\,c^2\,d+6\,B\,a\,b\,c\,d^2+6\,C\,a\,b\,c^2\,d\right )}{f}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (\frac {A\,a^2\,d^3}{2}-\frac {B\,a^2\,c^3}{2}-\frac {A\,b^2\,d^3}{2}+\frac {B\,b^2\,c^3}{2}-\frac {C\,a^2\,d^3}{2}+\frac {C\,b^2\,d^3}{2}-A\,a\,b\,c^3-B\,a\,b\,d^3+C\,a\,b\,c^3-\frac {3\,A\,a^2\,c^2\,d}{2}+\frac {3\,A\,b^2\,c^2\,d}{2}+\frac {3\,B\,a^2\,c\,d^2}{2}-\frac {3\,B\,b^2\,c\,d^2}{2}+\frac {3\,C\,a^2\,c^2\,d}{2}-\frac {3\,C\,b^2\,c^2\,d}{2}+3\,A\,a\,b\,c\,d^2+3\,B\,a\,b\,c^2\,d-3\,C\,a\,b\,c\,d^2\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (\frac {A\,b^2\,d^3}{4}+\frac {C\,a^2\,d^3}{4}-\frac {C\,b^2\,d^3}{4}+\frac {B\,a\,b\,d^3}{2}+\frac {3\,B\,b^2\,c\,d^2}{4}+\frac {3\,C\,b^2\,c^2\,d}{4}+\frac {3\,C\,a\,b\,c\,d^2}{2}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {B\,a^2\,d^3}{3}-\frac {b\,d^2\,\left (B\,b\,d+2\,C\,a\,d+3\,C\,b\,c\right )}{3}+\frac {C\,b^2\,c^3}{3}+\frac {2\,A\,a\,b\,d^3}{3}+A\,b^2\,c\,d^2+B\,b^2\,c^2\,d+C\,a^2\,c\,d^2+2\,B\,a\,b\,c\,d^2+2\,C\,a\,b\,c^2\,d\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {A\,a^2\,d^3}{2}-\frac {A\,b^2\,d^3}{2}+\frac {B\,b^2\,c^3}{2}-\frac {C\,a^2\,d^3}{2}+\frac {C\,b^2\,d^3}{2}-B\,a\,b\,d^3+C\,a\,b\,c^3+\frac {3\,A\,b^2\,c^2\,d}{2}+\frac {3\,B\,a^2\,c\,d^2}{2}-\frac {3\,B\,b^2\,c\,d^2}{2}+\frac {3\,C\,a^2\,c^2\,d}{2}-\frac {3\,C\,b^2\,c^2\,d}{2}+3\,A\,a\,b\,c\,d^2+3\,B\,a\,b\,c^2\,d-3\,C\,a\,b\,c\,d^2\right )}{f}+\frac {b\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (B\,b\,d+2\,C\,a\,d+3\,C\,b\,c\right )}{5\,f}+\frac {C\,b^2\,d^3\,{\mathrm {tan}\left (e+f\,x\right )}^6}{6\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^3*(A + B*tan(e + f*x) + C*tan(e + f*x)^2),x)

[Out]

x*(A*a^2*c^3 - A*b^2*c^3 + B*a^2*d^3 - C*a^2*c^3 - B*b^2*d^3 + C*b^2*c^3 + 2*A*a*b*d^3 - 2*B*a*b*c^3 - 2*C*a*b
*d^3 - 3*A*a^2*c*d^2 + 3*A*b^2*c*d^2 - 3*B*a^2*c^2*d + 3*B*b^2*c^2*d + 3*C*a^2*c*d^2 - 3*C*b^2*c*d^2 - 6*A*a*b
*c^2*d + 6*B*a*b*c*d^2 + 6*C*a*b*c^2*d) - (tan(e + f*x)*(B*a^2*d^3 - A*b^2*c^3 - b*d^2*(B*b*d + 2*C*a*d + 3*C*
b*c) - C*a^2*c^3 + C*b^2*c^3 + 2*A*a*b*d^3 - 2*B*a*b*c^3 - 3*A*a^2*c*d^2 + 3*A*b^2*c*d^2 - 3*B*a^2*c^2*d + 3*B
*b^2*c^2*d + 3*C*a^2*c*d^2 - 6*A*a*b*c^2*d + 6*B*a*b*c*d^2 + 6*C*a*b*c^2*d))/f - (log(tan(e + f*x)^2 + 1)*((A*
a^2*d^3)/2 - (B*a^2*c^3)/2 - (A*b^2*d^3)/2 + (B*b^2*c^3)/2 - (C*a^2*d^3)/2 + (C*b^2*d^3)/2 - A*a*b*c^3 - B*a*b
*d^3 + C*a*b*c^3 - (3*A*a^2*c^2*d)/2 + (3*A*b^2*c^2*d)/2 + (3*B*a^2*c*d^2)/2 - (3*B*b^2*c*d^2)/2 + (3*C*a^2*c^
2*d)/2 - (3*C*b^2*c^2*d)/2 + 3*A*a*b*c*d^2 + 3*B*a*b*c^2*d - 3*C*a*b*c*d^2))/f + (tan(e + f*x)^4*((A*b^2*d^3)/
4 + (C*a^2*d^3)/4 - (C*b^2*d^3)/4 + (B*a*b*d^3)/2 + (3*B*b^2*c*d^2)/4 + (3*C*b^2*c^2*d)/4 + (3*C*a*b*c*d^2)/2)
)/f + (tan(e + f*x)^3*((B*a^2*d^3)/3 - (b*d^2*(B*b*d + 2*C*a*d + 3*C*b*c))/3 + (C*b^2*c^3)/3 + (2*A*a*b*d^3)/3
+ A*b^2*c*d^2 + B*b^2*c^2*d + C*a^2*c*d^2 + 2*B*a*b*c*d^2 + 2*C*a*b*c^2*d))/f + (tan(e + f*x)^2*((A*a^2*d^3)/
2 - (A*b^2*d^3)/2 + (B*b^2*c^3)/2 - (C*a^2*d^3)/2 + (C*b^2*d^3)/2 - B*a*b*d^3 + C*a*b*c^3 + (3*A*b^2*c^2*d)/2
+ (3*B*a^2*c*d^2)/2 - (3*B*b^2*c*d^2)/2 + (3*C*a^2*c^2*d)/2 - (3*C*b^2*c^2*d)/2 + 3*A*a*b*c*d^2 + 3*B*a*b*c^2*
d - 3*C*a*b*c*d^2))/f + (b*d^2*tan(e + f*x)^5*(B*b*d + 2*C*a*d + 3*C*b*c))/(5*f) + (C*b^2*d^3*tan(e + f*x)^6)/
(6*f)

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